For any \(\pi \in \Pi \) , the eigenvalues of \(I-\gamma P^{\pi }\)  are contained in a circle centered at \((1, 0)\)  and with radius \(\gamma \)  in the complex plane.


The result follows directly from the fact that the eigenvalues of \(P^{\pi }\)  are contained in a circle centered at \((0,0)\)  and with radius 1 in the complex plane [1]}.

Finally, as depicted in Figure REF  for policy evaluation, the convergence rate of GMRES appears to be more robust against the discount factor compared to that of VI.
Algorithm REF  provides a pseudocode description of iGMRES-PI, where we have adopted the restarted version of GMRES to reduce the computational and storage complexity.
